Energy Eigenvalues of a Spin Half Particle Confined to a Circular Wire

Introduction:
When an uncharged spin half particle with magnetic moment μ is confined to move on a circular wire of radius a and subjected to a perpendicular magnetic field, it exhibits quantized energy levels. These energy eigenvalues can be derived using the principles of quantum mechanics.

Quantum Mechanical Treatment:
To determine the energy eigenvalues, we start by considering the Hamiltonian operator for the particle in the presence of a magnetic field. The Hamiltonian operator is given by:

H = -μ · B

Where μ is the magnetic moment of the particle and B is the magnetic field vector. In this case, B is perpendicular to the plane of the wire.

Zeeman Interaction:
The interaction between the magnetic moment and the magnetic field is known as the Zeeman interaction. It can be written as:

H = -μ · B = -γ · S · B

Where γ is the gyromagnetic ratio and S is the spin operator.

Energy Eigenvalues:
The energy eigenvalues can be obtained by solving the time-independent Schrödinger equation for the system:

Hψ = Eψ

Substituting the expression for the Hamiltonian, we get:

-γ · S · Bψ = Eψ

Since the particle is spin half, the spin operator S has two eigenvalues: +ħ/2 and -ħ/2. Therefore, we can write the above equation as two separate equations:

-γ · (+ħ/2) · Bψ = Eψ
-γ · (-ħ/2) · Bψ = Eψ

Simplifying the equations, we obtain:

E = ±γ · ħ/2 · B

These are the energy eigenvalues for the particle confined to move on a circular wire under the influence of a perpendicular magnetic field.

Conclusion:
In summary, when an uncharged spin half particle with magnetic moment μ is confined to move on a circular wire of radius a and subjected to a perpendicular magnetic field, the energy eigenvalues can be obtained by solving the time-independent Schrödinger equation. The energy eigenvalues are given by E = ±γ · ħ/2 · B, where γ is the gyromagnetic ratio and B is the magnetic field vector. These energy eigenvalues represent the quantized energy levels of the particle in this system.